An adaptive FEM with ITP approach for steady Schrödinger equation. Issue 1 (2nd January 2018)

• Record Type: Journal Article
• Authors: Kuang, Yang; Hu, Guanghui
• Is Part Of: International journal of computer mathematics. Volume 95:Issue 1(2018)
• Page Start: 187
• Page End: 201
• Publication Date: 2018-01-02
• Subjects: Schrödinger equation -- imaginary time propagation -- moving mesh method -- finite element method -- ground state; 65N30 -- 35Q40; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2017.1366463
• Languages: English
• ISSNs: 0020-7160

Abstract

ABSTRACT: In this paper, an adaptive numerical method is proposed for solving a 2D Schrödinger equation with an imaginary time propagation approach. The differential equation is first transferred via a Wick rotation to a real time-dependent equation, whose solution corresponds to the ground state of a given system when time approaches infinity. The temporal equation is then discretized spatially via a finite element method, and temporally utilizing a Crank–Nicolson scheme. A moving mesh strategy based on harmonic maps is considered to eliminate possible singular behaviour of the solution. Several linear and nonlinear examples are tested by using our method. The experiments demonstrate clearly that our method provides an effective way to locate the ground state of the equations through underlying eigenvalue problems.